Engineer Maths - Digital Communication
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paper_2.pdf
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Question 1
(a)
A digital source generates 32 symbols. One symbol has a probability of 0.5 and
another symbol has a probability of 0.1. The remaining 30 symbols each have a
probability of
. Compute the value for
and then the source entropy. Discuss
the steps involved in the above computation.
(5
marks)
(b)
A digital source generates 1024 equi-probable symbols. Use information theory
to determine the minimum number of bits required to represent each symbol. (5 marks)
(c) Let signal be a sinusoid defined as , where is
the amplitude, is the angular frequency and is the phase shift. Analyze the
signal by formulating an expression for its autocorrelation and power
spectral density (PSD).
(10 marks)
(d) It is required to support a data rate of 512 Mbps over a link with 128 MHz
bandwidth. Estimate the minimum signal-to-noise ratio (SNR) in dB scale
required to support the above data rate.
(5 marks)
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Question 2
(a) A digital communication system using multilevel signaling transmits one of the
32 possible levels in 0.4 ms. For the above system, calculate the number of bits
corresponding to each level, the baud rate and the bit rate.
(9 marks)
(b) Random processes and are zero-mean, independent and uncorrelated.
The power spectral density (PSD) of and are
and
respectively. Random process is obtained as
Therefore, the autocorrelation of is given by
, where and are the autocorrelation
functions of random processes and , respectively.
(i) The PSD of random process is denoted by Illustrate
with a sketch and a mathematical expression. Clearly label the amplitude
and frequency values in the sketch.
(6 marks)
(ii) Formulate an expression for the autocorrelation function using the
mathematical expression obtained in Question 2(b)(i).
(10 marks)
Question 3
(a) Construct the Polar RZ signalling waveform for the binary data sequence
0 1 1 0 1. Assume the bit duration to be 2 ms.
(5 marks)
(b) An analog signal is converted into PCM signal that is a binary polar NRZ line
code. The PCM quantizer has 64 steps and the overall equivalent system
transfer function is of the raised cosine-rolloff type with The PCM bit
rate is 12 kbps.
(i) Estimate the maximum bandwidth allowable for the analog signal.
(5 marks)
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(ii) Test whether a channel that is absolutely bandlimitted to 9 kHz can be
used to transmit the above PCM signal without introducing intersymbol
interference (ISI).
(5 marks)
(c) Signal is given by , where is the desired signal
and is the additive white Gaussian noise whose power spectral density
(PSD) is
.The signal is illustrated in Figure Q3(c). Assuming
that construct a matched filter to filter the signal such
that the signal-to-noise ratio at the filter output is maximized. Sketch the
impulse response of the matched filter. The sampling time should be such
that the matched filter is causal.
Figure Q3(c)
(10 marks)
Question 4
In a binary communication system, the receiver statistic corresponding to a polar
transmitted signal is corrupted by an additive noise which has triangular
distribution as given below. Figure Q4 graphically illustrates the triangular
distribution function .
.
Figure Q4
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The mean µ and variance for are given as follows:
.
The transmitted polar signals have values and , where is a positive
real number.
(a) Given that .
(i) Sketch the noise probability density function (PDF) and the
conditional PDFs for the receiver statistic , i.e., and .
Clearly label the sketch with appropriate values.
(10 marks)
(ii) Estimate the optimum detection threshold for the above system, if the
polar signals are equally likely. (Hint: )
(6 marks)
(iii) Rate the system performance by computing the bit error rate (BER)
corresponding to the optimum detection threshold obtained in
Question 4(a)(ii).
(2 marks)
(b) Given that Sketch the conditional PDFs
and . Use the conditional PDFs and propose a strategy to
detect the transmitted polar signals from the receiver statistic . Will there be
any detection errors?
(7 marks)
